The Basis of the Second Law of Thermodynamics in Quantum Field Theory D.W. Snoke1 , G. Liu Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA S. Girvin Department of Physics, Yale University, New Haven, CT 06520, USA Abstract The result that closed systems evolve toward equilibrium is derived entirely on the ba- sis of quantum field theory for a model system, without invoking any of the common extra- mathematical notions of particle trajectories, collapse of the wave function, measurement, or intrinsically stochastic processes. The equivalent of the Stosszahlansatz of classical statistical mechanics is found, and has important differences from the classical version. A novel result of the calculation is that interacting many-body systems in the infinite volume limit evolve toward diagonal states (states with loss of all phase information) on the the time scale of the interaction time. The connection with the onset of off-diagonal phase coherence in Bose condensates is discussed. 1. Introduction The controversy over the formulation of the Second Law of thermodynamics in terms of statistical mechanics of particles is well known [1]. Boltzmann presented his H-theorem as a way of connecting the deterministic, time-reversible dynamics of particles to the already well known, time-irreversible Second Law. This theorem, in Boltzmann’s formulation using classical mechanics, says that a many-particle system evolves irreversibly toward equilibrium via collision processes of the particles which can be calculated using known collision kinetics. Many scientists in his day found his arguments unpersuasive, however, partly because they involved notions of coarse graining which were ill-defined, and partly because the outcome of irreversibility seemed to contradict well-established ideas of time-reversibility such as Poincare cycles. Later on, statistical mechanics was given firmer footing on the basis of quantum mechanical states, and this approach is used in most modern textbooks [2]. Typically, the notion of entropy is introduced in terms of counting states, and the equilibrium distributions of quantum particles are deduced on the basis of particle statistics. The basic approach is the same as Boltzmann’s, using probability and statistics of particles, with the only difference being that the particles have well-defined quantum states. This gives the impression that the notion of indivisible particles is crucial for understanding statistical mechanics. arXiv:1112.3009v1 [cond-mat.stat-mech] 13 Dec 2011 1Email: [email protected], Phone: 412-624-9007, Fax: 412-624-9163 Preprint submitted to Elsevier October 15, 2018 In this article, we show that by using modern quantum field theory it is possible to deduce irreversible behavior for a closed system entirely on the basis of wave mechanics without ever invoking the concept of a random ensemble of particles. Starting with a nonequilibrium many- body wave function, an equivalent of the Boltzmann scattering integral can be written down, by which expectation values of the wave function describing the system can be evolved deter- ministically and irreversibly toward equilibrium in the thermodynamic limit. This method has already been used to successfully explain various real experiments, but we examine in detail the assumptions which are involved in justifying this approach. While one may interpret the many-body wave function as giving the probability of observa- tion of discrete particles, it is not necessary to do so in order to calculate any of the values we calculate here; the equilibration follows entirely from the mathematical structure of the contin- uous wave function. This calculation supports the philosophical view that the field of quantum field theory is “real.” By this we mean that the notion of particles is a subordinate concept of the field theory, useful in some circumstances and dispensable in others. In the case studied here of equilibration, the particle concept is dispensable. In general, particles are entirely deducible from the field theory as eigenstates (resonances) of the field. The many-body wave function is therefore a more fundamental physical description of the system. A surprising result of the calculation here is that systems generally evolve toward diago- nal states (many-body states with no phase coherence between different single-particle quantum states) on time scales comparable to the collision time. Thus, the calculation shows a type of “collapse” of the wavefunction that comes entirely from the mathematics of quantum field the- ory without need for invoking external observers or other, more esoteric concepts. Several other works have addressed the evolution of a quantum mechanical system toward equilibrium, notably Refs. [3] and [4]. These calculations have typically taken the approach of using a “bath” or “environment” which interacts with a smaller subsystem; the bath is assumed to already be in equilibrium. In the present work, we treat the entire closed system as one system, with the same equations of evolution for all parts, and all of which can be far from equilibrium. The essential puzzle we face is that in this case, quantum evolution is unitary and hence the von Neumann entropy of a closed system ought to be a constant of the motion. This situation has been studied by Gellman and Hartle [5], who used coarse-graining ideas to study the time evolution of quantum systems and the cross-over to classical irreversible evolution. Polkovnikov [6] has introduced the concept of the microscopic “diagonal entropy,” which is the von Neumann entropy computed ignoring the contributions from the off-diagonal parts of the density matrix. He has shown that the diagonal entropy has all the properties expected of the thermodynamic entropy. We will present here a derivation of the quantum Boltzmann equation which yields irreversible time evolution and show that the quantum coherences represented by simple off- diagonal elements of the density matrix do indeed (generically) decay towards zero. As shown in Appendix A, an odd result of the Polkovnikov definition of diagonal entropy is that the entropy of a boson system near condensation decreases for some period of time. A generalized H-theorem can still be derived with a revision of the definition of entropy that also involves only diagonal elements, but which has less obvious connection to the von Neumann entropy. A key concept in modern quantum information theory that helps us resolve the paradox de- scribed above is that of entanglement entropy [7, 8]. A strange feature of quantum mechanics is that a system can be in a pure quantum state with zero entropy even though each of its sub- systems has finite entropy. This is because the entanglement between the different subsystems gives non-classical correlations which make a negative contribution to the entropy of the overall system. For example, the reduced density matrix for a portion of a quantum system in its ground 2 state obtained by tracing over the other parts of the system describes a mixed state with finite energy fluctuations and finite entropy [8]. Coarse graining, in which we trace over short distance degrees of freedom to obtain a long-wavelength description [5], can thus lead to the appearance of decoherence, classicality, and entropy. We discuss below how apparently irreversible decoherence naturally arises within the ap- proximations made in deriving the quantum Boltzmann equation. A new result here is that the average values of the off-diagonal elements of the generalized density matrix, which hold the in- formation about the entanglement, all tend exponentially quickly toward zero. In the low-density limit, in which both bosons and fermions recover classical statistics, the ratio of the off-diagonal elements to the diagonal elements strictly vanishes. Thus, while the total von Neumann entropy does stay constant, the entropy in the off-diagonal terms becomes spread over so many possible entanglements that it becomes inaccessible to experiments. 2. Derivation of the Quantum Boltzmann Equation The equation which describes the evolution of a many-body system toward equilibrium is commonly known as the quantum Boltzmann equation; to be accurate, the term “quantum Boltzmann equation” should be used for an equation which describes both the real-space and momentum-space evolution of a many-body system, while the term “Fokker-Planck equation” should be used for an equation for the evolution in momentum space only, but in the case of a homogeneous gas, these are the same, and the two terms are often used interchangeably. The full classical Boltzmann equation is usually assumed to include terms for the spatial drift of the particles (see, e.g., Ref. [9], Section 5.9). Therefore we will use the term “classical Boltzmann integral” when referring below to the analogous classical calculation for the evolution of the particle distribution function. A basic result of quantum field theory which we will use here is that a quantum field can be written in terms of second-quantization operators which obey the commutation relation y y y [ak; ak] = akak − akak = 1 (1) for bosons, and the anticommutation relation y y y fak; akg = akak + akak = 1 (2) for fermions. In a homogeneous system, the subscript corresponds to a wave vector ~k. (We drop the vector notation on subscripts everywhere in this paper.) The mathematics of these operators goes back to Dirac [10] and is presented in many recent textbooks such as Refs. [11] and [12]. y These operators naturally lead to the language of particles, because the operator ak acting on a ˆ y wave state increases the eigenvalue of the operator Nk = akak by 1 and the operator ak decreases the eigenvalue of Nˆ k by 1, making it natural to treat Nˆ k as giving an integer number of particles, y and ak as the “creation” operator and ak as the “destruction” operator for single particles. More y fundamentally, the ak and ak operators represent the complex amplitude of a wave of the quantum field in mode ~k, and Nˆ k represents its intensity.